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Derivations, Gradings, Actions of Algebraic Groups, and Codimension Growth of Polynomial Identities

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Abstract

Suppose a finite dimensional semisimple Lie algebra \(\mathfrak g\) acts by derivations on a finite dimensional associative or Lie algebra A over a field of characteristic 0. We prove the \(\mathfrak g\)-invariant analogs of Wedderburn—Mal’cev and Levi theorems, and the analog of Amitsur’s conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of A. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur’s conjecture for finite dimensional associative algebras with an action of a reductive affine algebraic group by automorphisms and anti-automorphisms or graded by an arbitrary Abelian group. In addition, we provide criteria for G-, H- and graded simplicity in terms of codimensions.

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References

  1. Aljadeff, E., Giambruno, A.: Multialternating graded polynomials and growth of polynomial identities. Proc. Am. Math. Soc. (to appear)

  2. Aljadeff, E., Giambruno, A., La Mattina, D.: Graded polynomial identities and exponential growth. J. Reine Angew. Math., 650, 83–100 (2011)

    MATH  MathSciNet  Google Scholar 

  3. Bakhturin, Yu.A.: Identical Relations in Lie Algebras. VNU Science Press, Utrecht (1987)

    MATH  Google Scholar 

  4. Bahturin, Yu.A., Giambruno, A., Zaicev, M.V.: G-identities on associative algebras. Proc. Am. Math. Soc. 127(1), 63–69 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bahturin, Yu.A., Linchenko, V.: Identities of algebras with actions of Hopf algebras. J. Algebra 202(2), 634–654 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bahturin, Yu.A., Zaicev, M.V.: Identities of graded algebras. J. Algebra, 205, 1–12 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bahturin, Yu.A., Zaicev, M.V.: Identities of graded algebras and codimension growth. Trans. Am. Math. Soc. 356(10), 3939–3950 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bakhturin, Yu.A., Zaĭtsev, M.V. Sehgal, S.K.: G-identities of non-associative algebras. Sb.: Math. 190(11), 1559–1570 (1999)

    MATH  MathSciNet  Google Scholar 

  9. Berele, A.: Cocharacter sequences for algebras with Hopf algebra actions. J. Algebra, 185, 869–885 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chuang, C.-L., Lee, T.-K.: q-skew derivations and polynomial identities. Manuscripta Math. 116, 229–243 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dixmier, J.: Enveloping Algebras. Amsterdam, North-Holland (1977)

  12. Dăscălescu, S., Năstăsescu, C., Raianu, Ş.: Hopf algebras: an introduction. Marcel Dekker, Inc., New York (2001)

    Google Scholar 

  13. De Filippis, V.: Power cocentralizing generalized derivations on prime rings. Proc. Indian Acad. Sci., Math. Sci. 120(3), 285–297 2010

    Article  MATH  MathSciNet  Google Scholar 

  14. Drensky, V.S.: Free Algebras and PI-algebras: Graduate Course in Algebra. Springer-Verlag, Singapore (2000)

    Google Scholar 

  15. Giambruno, A., La Mattina, D.: Graded polynomial identities and codimensions: computing the exponential growth. Adv. Math. 225, 859–881 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Giambruno, A., Shestakov, I.P., Zaicev, M.V.: Finite-dimensional non-associative algebras and codimension growth. Adv. Appl. Math. 47, 125–139 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Giambruno, A., Zaicev, M.V.: Polynomial identities and asymptotic methods. In: AMS Mathematical Surveys and Monographs, vol. 122. Providence, R.I. (2005)

  18. Gordienko, A.S.: Codimensions of generalized polynomial identities. Sb.: Math. 201(2), 235–251 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Gordienko, A.S.: Codimensions of polynomial identities of representations of Lie algebras. Proc. Am. Math. Soc. (to appear)

  20. Gordienko, A.S.: Graded polynomial identities, group actions, and exponential growth of Lie algebras. J. Algebra 367, 26–53 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gordienko, A.S.: Amitsur’s conjecture for associative algebras with a generalized Hopf action. J. Pure Appl. Alg. (to appear)

  22. Gordienko, A.S.: Structure of H-(co)module Lie algebras. J. Lie Theory (to appear).

  23. Gordienko, A.S.: Amitsur’s conjecture for polynomial H-identities of H-module Lie algebras. Tran. Amer. Math. Soc. (to appear)

  24. Goto, M., Grosshans, F.: Semisimple Lie Algebras. Marcel Dekker, New York and Basel (1978)

    MATH  Google Scholar 

  25. Hochschild, G.: Basic theory of algebraic groups and Lie algebras. Graduate Texts in Mathematics, vol. 75. Springer-Verlag, New York (1981)

    Book  MATH  Google Scholar 

  26. Humphreys, J.E.: Linear algebraic groups. Springer-Verlag, New-York (1975)

    Book  MATH  Google Scholar 

  27. Jacobson, N.: Lie Algebras. Interscience Publishers, New York–London (1962)

    MATH  Google Scholar 

  28. Kharchenko, V.K.: Differential identities of semiprime rings. Algebra Log. 18, 86–119 (1979)

    Article  MathSciNet  Google Scholar 

  29. Linchenko, V.: Identities of Lie algebras with actions of Hopf algebras. Commun. Algebra 25(10), 3179–3187 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Mishchenko, S.P., Verevkin, A.B., Zaitsev, M.V.: A sufficient condition for coincidence of lower and upper exponents of the variety of linear algebras. Mosc. Univ. Math. Bull. 66(2), 86–89 (2011)

    Article  MathSciNet  Google Scholar 

  31. Montgomery, S.: Hopf algebras and their actions on rings. In: CBMS Lecture Notes, vol. 82. Amer. Math. Soc. Providence, RI (1993)

  32. Pagon, D., Repovš, D., Zaicev, M.V.: Group gradings on finite dimensional Lie algebras. Alg. Colloq. (to appear)

  33. Sharma, D.K., Dhara, B.: Engel type identities with derivations for right ideals in prime rings. Mediterr. J. Math. 3, 15–29 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sidorov, A.V.: Splitting of the radical in finite-dimensional H-module algebras. Algebra Log. 28(3, 324–336 (1986)

    MathSciNet  Google Scholar 

  35. Ştefan, D., Van Oystaeyen, F.: The Wedderburn—Malcev theorem for comodule algebras. Comm. Algebra 27(8), 3569–3581 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sweedler, M.: Hopf Algebras. Benjamin, W.A., Inc., New York (1969)

    Google Scholar 

  37. Taft, E.J.: Invariant Wedderburn factors. Illinois J. Math. 1, 565–573 (1957)

    MATH  MathSciNet  Google Scholar 

  38. Volichenko, I.B.: Varieties of Lie algebras with identity [[X 1, X 2, X 3], [X 4, X 5, X 6]] = 0 over a field of characteristic zero. (Russian) Sibirsk. Mat. Zh. 25(3), 40–54 (1984)

    MathSciNet  Google Scholar 

  39. Zaitsev, M.V.: Integrality of exponents of growth of identities of finite-dimensional Lie algebras. Izv. Math. 66, 463–487 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  40. Zaicev, M.V., Mishchenko, S.P.: An example of a variety of Lie algebras with a fractional exponent. J. Math. Sci. (New York) 93(6), 977–982 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Free University of Brussels, Brussel, Belgium

    A. S. Gordienko

  2. Memorial University of Newfoundland, St. John’s, NL, Canada

    M. V. Kochetov

Authors
  1. A. S. Gordienko
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  2. M. V. Kochetov
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Correspondence to A. S. Gordienko.

Additional information

The first author is supported by FWO Vlaanderen Pegasus Marie Curie post doctoral fellowship (Belgium). The second author is supported by by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Discovery Grant # 341792-07.

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Gordienko, A.S., Kochetov, M.V. Derivations, Gradings, Actions of Algebraic Groups, and Codimension Growth of Polynomial Identities. Algebr Represent Theor 17, 539–563 (2014). https://doi.org/10.1007/s10468-013-9409-z

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  • DOI: https://doi.org/10.1007/s10468-013-9409-z

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